[math-fun] Cut, flip, replace, repeat
In the September 2014 issue of Math Horizons (page 8), Stan Wagon poses the following problem, which he says originated as a problem in the 1968 Moscow Mathematical Olympiad: A round cake has icing only on top. Cut out a piece in the usual shape (a sector of the circle), turn it upside down, and replace it to restore roundness. Repeat with the next piece; that is, move counterclockwise, cut out a piece with the same central angle, flip it, and replace it. Continue this process. If a piece has a knife cut from a previous iteration, ignore the cut and flip the piece as if it was solid. For pieces with 45 degree central angles, it takes 16 flips to return all icing to the top of the cake; if 180 degrees is used, it takes only four flips. How many flips does it take when the central angle is 181 degrees? Pete Winkler poses a similar problem in Mathematical Mind-Benders, which he says "was passed to me by French graduate student Thierry Mora, who heard it from his prep-school teacher Thomas Laorgue" (who didn't invent the problem and didn't know its origin). Does anyone have access to the 1968 Moscow Olympiad problems? If so, can he/she provide an English translation of the original problem? I'd be especially interested in knowing who is credited with inventing the problem (though often this kind of information is hard to dig up). Jim Propp
Actually, I found it at http://andrej.fizika.org/ostalo/gimnazija/math/ruske_olimpijade/11a-olym-1.p... : 31.2.8.3*. A round pie is cut by a special cutter that cuts off a fixed sector of the angle measure α, turns this sector upside down, and then inserts back; after that the whole pie is rotated through an angle of β. Given β < α < 180 degrees, prove that after a finite number of such operations (the beginning of the first and the second operations are shown on Fig. 67) every point of the pie will return to its initial place. Jim Propp Jim Propp On Thu, Sep 11, 2014 at 2:58 PM, James Propp <jamespropp@gmail.com> wrote:
In the September 2014 issue of Math Horizons (page 8), Stan Wagon poses the following problem, which he says originated as a problem in the 1968 Moscow Mathematical Olympiad:
A round cake has icing only on top. Cut out a piece in the usual shape (a sector of the circle), turn it upside down, and replace it to restore roundness. Repeat with the next piece; that is, move counterclockwise, cut out a piece with the same central angle, flip it, and replace it. Continue this process. If a piece has a knife cut from a previous iteration, ignore the cut and flip the piece as if it was solid. For pieces with 45 degree central angles, it takes 16 flips to return all icing to the top of the cake; if 180 degrees is used, it takes only four flips. How many flips does it take when the central angle is 181 degrees?
Pete Winkler poses a similar problem in Mathematical Mind-Benders, which he says "was passed to me by French graduate student Thierry Mora, who heard it from his prep-school teacher Thomas Laorgue" (who didn't invent the problem and didn't know its origin).
Does anyone have access to the 1968 Moscow Olympiad problems? If so, can he/she provide an English translation of the original problem?
I'd be especially interested in knowing who is credited with inventing the problem (though often this kind of information is hard to dig up).
Jim Propp
Are α and β allowed to be irrational numbers? Brent Meeker On 9/11/2014 12:12 PM, James Propp wrote:
Actually, I found it at http://andrej.fizika.org/ostalo/gimnazija/math/ruske_olimpijade/11a-olym-1.p... :
31.2.8.3*. A round pie is cut by a special cutter that cuts off a fixed sector of the angle measure α, turns this sector upside down, and then inserts back; after that the whole pie is rotated through an angle of β. Given β < α < 180 degrees, prove that after a finite number of such operations (the beginning of the first and the second operations are shown on Fig. 67) every point of the pie will return to its initial place.
Jim Propp
Jim Propp
On Thu, Sep 11, 2014 at 2:58 PM, James Propp <jamespropp@gmail.com> wrote:
In the September 2014 issue of Math Horizons (page 8), Stan Wagon poses the following problem, which he says originated as a problem in the 1968 Moscow Mathematical Olympiad:
A round cake has icing only on top. Cut out a piece in the usual shape (a sector of the circle), turn it upside down, and replace it to restore roundness. Repeat with the next piece; that is, move counterclockwise, cut out a piece with the same central angle, flip it, and replace it. Continue this process. If a piece has a knife cut from a previous iteration, ignore the cut and flip the piece as if it was solid. For pieces with 45 degree central angles, it takes 16 flips to return all icing to the top of the cake; if 180 degrees is used, it takes only four flips. How many flips does it take when the central angle is 181 degrees?
Pete Winkler poses a similar problem in Mathematical Mind-Benders, which he says "was passed to me by French graduate student Thierry Mora, who heard it from his prep-school teacher Thomas Laorgue" (who didn't invent the problem and didn't know its origin).
Does anyone have access to the 1968 Moscow Olympiad problems? If so, can he/she provide an English translation of the original problem?
I'd be especially interested in knowing who is credited with inventing the problem (though often this kind of information is hard to dig up).
Jim Propp
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I'm confused by issues about α and β and the meaning of "such operations". Are we supposed to be given arbitrary β < α < 180 degrees and then alternate them? Or is the question whether β < α < 180 can be chosen so that alternation returns the pie to its original state after finitely many operations using α and β alternately? (Assuming a self=healing pie.) Or ??? --Dan << A round pie is cut by a special cutter that cuts off a fixed sector of the angle measure α, turns this sector upside down, and then inserts back; after that the whole pie is rotated through an angle of β. Given β < α < 180 degrees, prove that after a finite number of such operations (the beginning of the first and the second operations are shown on Fig. 67) every point of the pie will return to its initial place.
Dan asks: "Are we supposed to be given arbitrary β < α < 180 degrees and then alternate them?" Yes. The surprise is that regardless of how α and β are chosen, alternation returns the pie to its original state after finitely many operations using α and β alternately. Jim Propp
Thanks. I know I saw a very similar problem somewhere in the past 10 years. Maybe in the Monthly? Though I'm pretty sure it was about a cake, not a pie. --Dan On Sep 11, 2014, at 1:28 PM, James Propp <jamespropp@gmail.com> wrote:
Dan asks: "Are we supposed to be given arbitrary β < α < 180 degrees and then alternate them?"
Yes.
The surprise is that regardless of how α and β are chosen, alternation returns the pie to its original state after finitely many operations using α and β alternately.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Brent asks: Are α and β allowed to be irrational numbers? Yes. At this point, some people object "But then the claim is false, because you never cut the cake in the same place twice!" One can smirk and say something like "That's what a lot of smart people say" (that's the sort of approach Pete Winkler usually takes). Or one can point out that the pieces get flipped, so that the non-repeating nature of the set of fractional parts of an irrational number, while true, becomes irrelevant. Jim On Thu, Sep 11, 2014 at 3:55 PM, meekerdb <meekerdb@verizon.net> wrote:
Are α and β allowed to be irrational numbers?
Brent Meeker
On 9/11/2014 12:12 PM, James Propp wrote:
Actually, I found it at http://andrej.fizika.org/ostalo/gimnazija/math/ruske_ olimpijade/11a-olym-1.pdf :
31.2.8.3*. A round pie is cut by a special cutter that cuts off a fixed sector of the angle measure α, turns this sector upside down, and then inserts back; after that the whole pie is rotated through an angle of β. Given β < α < 180 degrees, prove that after a finite number of such operations (the beginning of the first and the second operations are shown on Fig. 67) every point of the pie will return to its initial place.
Jim Propp
Jim Propp
On Thu, Sep 11, 2014 at 2:58 PM, James Propp <jamespropp@gmail.com> wrote:
In the September 2014 issue of Math Horizons (page 8), Stan Wagon poses
the following problem, which he says originated as a problem in the 1968 Moscow Mathematical Olympiad:
A round cake has icing only on top. Cut out a piece in the usual shape (a sector of the circle), turn it upside down, and replace it to restore roundness. Repeat with the next piece; that is, move counterclockwise, cut out a piece with the same central angle, flip it, and replace it. Continue this process. If a piece has a knife cut from a previous iteration, ignore the cut and flip the piece as if it was solid. For pieces with 45 degree central angles, it takes 16 flips to return all icing to the top of the cake; if 180 degrees is used, it takes only four flips. How many flips does it take when the central angle is 181 degrees?
Pete Winkler poses a similar problem in Mathematical Mind-Benders, which he says "was passed to me by French graduate student Thierry Mora, who heard it from his prep-school teacher Thomas Laorgue" (who didn't invent the problem and didn't know its origin).
Does anyone have access to the 1968 Moscow Olympiad problems? If so, can he/she provide an English translation of the original problem?
I'd be especially interested in knowing who is credited with inventing the problem (though often this kind of information is hard to dig up).
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Dan Asimov -
James Propp -
meekerdb